Mathoverflow 339137

Let P(x), Q(x) ∈ ℝ[x] be two monic polynomials with non-negative coefficients. If R(x) = P(x)Q(x) is a 0,1 polynomial (coefficients only from 0,1), then P(x) and Q(x) are also 0, 1 polynomials.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-mathoverflow-339137,
  title        = {Mathoverflow 339137},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/mathoverflow-339137}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $P (x), Q (x) \in R [x]$ be two monic polynomials with non-negative coefficients. If $R (x) = P (x) Q (x)$ is a $0, 1$ polynomial (coefficients only from ${0, 1}$ ), then $P (x)$ and $Q (x)$ are also $0, 1$ polynomials.

Why do polynomials with coefficients 0,1 like to have only factors with 0,1 coefficients?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Mathoverflow.«339137».

theorem mathoverflow_339137 (P Q R : ℝ[X]) (hP: P.Monic) (hQ : Q.Monic)
    (hp : ∀ c ∈ P.coeffs, 0 ≤ c) (hq : ∀ c ∈ Q.coeffs, 0 ≤ c)
    (h : R = P * Q) (hR : IsZeroOne R) :
    IsZeroOne P ∧ IsZeroOne Q

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

mathoverflow/339137 asked by user Sil

Source and licence

Imported from Formal Conjectures (MathOverflow), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.